On Dec 22, 8:55 am, WM <mueck...@rz.fh-augsburg.de> wrote:> On 22 Dez., 00:34, William Hughes <wpihug...@gmail.com> wrote:>> > > Yes it is definable. It has been defined. Nevertheless it does not> > > exist, because the sets do not exist.>> > It is definable which means that WM can use it to prove that the> > bijection exists, but it does not exist.>> The bijection of all finite words with all natural numbers has been> defined in binary:

However, the bijection you need is all definitons with a subsetof the natural numbers. And there is no way to define this subset.If you put restrictions on the 0/1 sequences you allow to exist youput restrictions on the subsets you allow to exist. Note, thatsubcountabledoes not mean countable.

>> 0> 1> 00> 01> 10> 11> 000> and so on.>> From this definition the natural number belonging to any desired> finite word can be obtained. It can easily be translated into any> other language. Or do you need some help?>> Nevertheless there is no set of all natural numbers and no set of all> infinite words.>> It is the same with pi. The (potentially) infinite string of digits of> pi can be defined. In fact there are (potentially) infinitely many> definitions. Nevertheless there is no actually infinite string> expressing pi.>> Yes, I know that is not easy to understand. That's why so many> mathematicians have gone astray.>> Regards, WM